Have you realised the result of each of [tex]a_p=\sqrt{(p^2+p)}-p[/tex]

Plug in the numbers 1 to 15? What do you get? What can you therefore tentatively assume about the limit? Now plug in the numbers from 15 to 30, what do you get? Again, what tentative assumption can you make? now 100 to 150?

The numerical exercise can give moral justification to the answer.
However, to actually prove it, try the method of "conjugates". What is the conjugate of the function? Multiply and divide by the conjugate. The limit works out after some algebra.